Let \((v_k) := (u_{i_k})\)  be a subsequence of \((u_i)\) , such that
\(\Vert  v_k\Vert _\varphi \leqslant 2^{-k}.\) 
Let \(\Gamma \subset \Gamma _{rect}(\Omega )\)  be the family of curves \(\gamma \) , such that \(\int _\gamma v_k \,ds \nrightarrow 0\)  as \(k \rightarrow \infty \) .
For every \(j \in \mathbb {N}\) , let
\(w_j:= \sum _{k=1}^j v_k.\) 
Since every \(v_k\)  is a non-negative Borel function, every \(w_j\)  is also a non-negative Borel function.
Since the sequence \((w_j(x))\)  is increasing for every \(x\in \Omega \) , the limit \(w(x) := \lim _{j\rightarrow \infty }w_j(x)\)  (possibly \(\infty \) ) exists.
By [1]}, if \(j<m\) , then
\(\Vert w_m-w_j\Vert _\varphi =\Big {\Vert } \sum _{k=j+1}^m v_k \Big {\Vert }_\varphi \lesssim \sum _{k=j+1}^m \Vert  v_k \Vert _\varphi \leqslant \sum _{k=j+1}^m 2^{-k}< 2^{-j},\) 
which implies that \((w_j)\)  is a Cauchy sequence in \(L^\varphi (\Omega )\) .
By Lemma REF , \(w\)  is the limit of \((w_j)\)  in \(L^\varphi (\Omega )\) , which implies that \(w \in L^\varphi (\Omega )\) , and therefore \(\Vert  w \Vert _\varphi < \infty \) .
Suppose now that \(\gamma \in \Gamma \) .
Then
\(\int _\gamma w \,ds= \sum _{k=1}^\infty \int _\gamma v_k \,ds= \infty ,\) 
because \(\sum _{k=1}^\infty \int _\gamma v_k \,ds < \infty \)  would imply that \(\lim _{k\rightarrow \infty }\int _\gamma v_k \,ds = 0\) .
Thus \(w/m\)  is \(\Gamma \) -admissible for every \(m\in \mathbb {N}\) .
Since \(\lim _{m\rightarrow \infty } \Vert w/m\Vert _\varphi = \lim _{m\rightarrow \infty } \Vert w\Vert _\varphi / m = 0\) , we have \(\operatorname{M_{\varphi }}(\Gamma ) = 0\) .
